"Identical vertices" are usually taken to mean that for any two vertices, there must be an isometry of the entire solid that takes one vertex to the other. Sometimes it is instead only required that the faces that meet at one vertex are related isometrically to the faces that meet at the other. This difference in definitions controls whether the elongated square gyrobicupola (pseudo-rhombicuboctahedron) is considered an Archimedean solid or a Johnson solid: it is the unique convex polyhedron that has regular polygons meeting in the same way at each vertex, but that does not have a global symmetry taking every vertex to every other vertex. Based on its existence, Branko Grünbaum (2009) has suggested a terminological distinction in which an Archimedean solid is defined as having the same vertex figure at each vertex (including the elongated square gyrobicupola) while a uniform polyhedron is defined as having each vertex symmetric to each other vertex (excluding the gyrobicupola).

Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville.[1]

Here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of (4,6,8) means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).

The snub cube and snub dodecahedron are known as chiral, as they come in a left-handed (Latin: levomorph or laevomorph) form and right-handed (Latin: dextromorph) form. When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain chemical compounds).

The Archimedeans solids can be constructed as generator positions in a kaleidoscope.

The different Archimedean and Platonic solids can be related to each other using just several general constructions. Starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated (see table below), different Platonic and Archimedean (and other) solids can be created. Expansion or cantellation involves moving each face away from the center (by the same distance so as to preserve the symmetry of the Platonic solid) and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles. The last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges but with a particular ratio between corner and edge truncation.

Note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron. Also, in part due to self-duality of the tetrahedron, only one Archimedean solid has only tetrahedral symmetry.